On the Remes Algorithm for Rational Approximations


  • Husam L Saad College of Science, University of Basrah, Basrah
  • Noori Yasir Abdul Hassan College of Education for Pure Science, University of Basrah, Basrah




Minimax approximation, Rational functions, Remes algorithm, Nonlinear system of leveling Equations, The dual monomial Vandermond system, The leveled reference error.


This paper is concerned with the minimax approximation of a discrete data set by rational functions. The second algorithm of Remes is applied. A crucial stage of this algorithm is solving the nonlinear system of leveling equations. In this paper, we will give a new approach for this purpose. In this approach, no initial guesses are required. Illustrative numerical example is presented.


Download data is not yet available.

Author Biography

Noori Yasir Abdul Hassan, College of Education for Pure Science, University of Basrah, Basrah

Department of Mathematics


[1] Barrodale, I. and Mason, J. C. (1970)," Two Simple Algorithms for Discrete Rational Approximation", Maths. of Comp., 26, 877-891.
[2] Barrodale, I. and Philips, C. (1975), "Algorithm 495: Solution of an Overdetermined System of Linear Equations in the Chebyshev Norm", ACM Trans. Maths. Software, 1, 264-270.
[3] BjÓ§rck, A. and Pereyra, V. (1970), "Solution of Vandermonde System of Equations", Maths. of Comp., 24, 893-903.
[4] Cheney, E. W. (1966), "Introduction to Approximation Theory", McGraw-Hill.
[5] Chun, C. and Ham, Y. (2008), "Some Fourth-Order Modifications of Newton's Method", Applied Mathematics and Computation",197, 654-658.
[6] Davis, P. J. (1975), "Interpolation and Approximation", Dover Publications, New York.
[7] Higham, N. J. (1988), "Fast Solution of Vandermonde-like Systems Involving Orthogonal Polynomials", J. Numer.
Math., 8, 473-486.
[8] Kaufman, J. R., Leeming, D. G. and Taylor, G. D. (1980), "A Combined Remes-Differential Correction Algorithm for Rational Approximation", Experimental Results, J. Comp. And Maths. with Appl., 6,155-166.
[9] Lee, C. M. and Roberts, F. D. (1973), "A Comparison of Algorithm for Rational Approximation", Math. of Comp., 26, 111-120.
[10] Meinardus, G. (1967), "Approximation of Functions: Theory and Numerical Methods", Springer, Heidelberg.
[11] Mhaskar, H. N. and Pai, D. V. (2000), "Fundamentals of Approximation Theory", Narosa Publishing House, New Delhi. [12] Pachá½¹n, R. and Trefethen, L.N. (2009), "Barycentric-Remez Algorithms for Test Polynomial Approximation in the Chebfun System", BIT Numer. Math., 49, 721-741.
[13] Powell, M. J. D. (1981), "Approximation Theory and Methods", Cambridge University Press, Cambridge, UK.
[14] Remez, E.Y. (1969), "Fundamentals of Numerical Methods for Chebyshev Approximations", Naukova Dumka, Kiev.
[15] Rice, J. R. (1964), "The Approximation of Functions (Vol. 1)", Addison-Wesley.
[16] Steffens, K. G. (2006), "The History of Approximation Theory: From Euler to Bernstein", Birkhäuser, Boston.
[17] Watson, G. A. (1980), "Approximation Theory and Numerical Methods", John Wiley and Sons.




How to Cite

Saad, H. L., & Hassan, N. Y. A. (2016). On the Remes Algorithm for Rational Approximations. JOURNAL OF ADVANCES IN MATHEMATICS, 12(10), 6733–6738. https://doi.org/10.24297/jam.v12i10.106